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Communication Melting in Graphs and Complex Networks

Complex networks are the representative graphs of interactions in many complex systems. Usually, these interactions are abstractions of the communication/diffusion channels between the units of the system. Real complex networks, e.g. traffic networks, reveal different operation phases governed by the dynamical stress of the system. Here we show how, communicability, a topological descriptor that reveals the efficiency of the network functionality in terms of these diffusive paths, could be used to reveal the transitions mentioned. By considering a vibrational model of nodes and edges in a graph/network at a given temperature (stress), we show that the communicability function plays the role of the thermal Green's function of a network of harmonic oscillators. After, we prove analytically the existence of a universal phase transition in the communicability structure of every simple graph. This transition resembles the melting process occurring in solids. For instance, regular-like graphs resembling crystals, melts at lower temperatures and display a sharper transition between connected to disconnected structures than the random spatial graphs, which resemble amorphous solids. Finally, we study computationally this graph melting process in some real-world networks and observe that the rate of melting of graphs changes either as an exponential or as a power-law with the inverse temperature. At the local level we discover that the main driver for node melting is the eigenvector centrality of the corresponding node, particularly when the critical value of the inverse temperature approaches zero. These universal results sheds light on many dynamical diffusive-like processes on networks that present transitions as traffic jams, communication lost or failure cascades.


Introduction
The use of graphs and networks to represent many physical, biological, social and engineering systems has triggered their relevance as an object of study in applied mathematics [9,10,25].
One of the main goals of these studies is to understand the robustness of these networks to the external stresses to which they are constantly submitted to. In this sense, the study of melting processes of graphs and networks can bring some new lights on this important area of applied research. A network can be considered as a general system of balls and springs submerged into a thermal bath at a given inverse temperature β = (k B T ) −1 where k B is a constant [13].
Here the thermal bath represents the external stress to which the system is submitted to and β represents a weight applied to every edge of the graph (see Preliminaries for details). When the external stress is too strong, β → 0, the graph is fully disconnected indicating that no transfer of "information" is possible between any pair of nodes in the network. The capacity of a node to transmit a perturbation at a given β to another node is quantified by the thermal Green's function of the network [13]. This function is better known in the literature as the communicability function of a graph [12,13]. It has found many applications in the analysis of real-world networks, such as in detecting changes in the contralesional hemisphere following strokes in humans [8], in the detection of symptoms of multiple sclerosis [18], in the study of variants of epilepsy [6], in prediction of abnormal brain states [16], in early detection of Alzheimer's disease [22], in prediction of functional protein complexes [20], in the analysis of genetic diseases [5], in the optimization of wireless networks [7], in the evolution of granular materials [32], in the classification of grass pollen [24] and vegetation patterns [23], and in the identification of the transcription factor critically involved with self-renewal of undifferentiated embryonic stem cells [21], to mention just a few of recent findings.
Melting-the phase transition in which a solid is transformed into a liquid-is a fundamental physical process of elements, substances and materials, which results from the application of heat or pressure to the substance [4,2]. One of the most successful criteria for explaining melting at the microscopic level was developed by Lindemann in 1910 [19]. According to Lindemann criterion [19,17], melting is caused by vibration instability in the crystal lattice, which eventually makes that the amplitude of vibration becomes so large that the atoms collide with their nearest neighbors, disturbing them and initiating the melting. Then, every substance is characterized by a melting point, which is the temperature at which such process starts. A crystal can be represented by a regular lattice [27] in which atoms are the nodes and interactions between atoms are the edges of a simple graph. It is then easy to set up a vibrational model on this graph by considering it as a ball-and-spring system and studying the change of state in it as a result of raising the temperature using the Lindemann criterion [19]. Many granular materials are nowadays studied by using graph-theoretic methods [26].
Thus, such approach to study melting using graphs is of great importance in this area of research. However, the most important question that immediately emerges here is whether we can generalize such theoretical framework to consider any simple graph. That is, can we use the physical metaphor of "melting" for general graphs and networks?
In this work we consider a Lindemann-like model for the melting of graphs and networks.
That is, we consider a vibrational model of nodes in a network based on the communicability function. Then, we prove analytically the existence of a universal phase transition in the communicability structure of every simple graph, which resembles the melting process occurring in substances. We discovered that, similar to crystalline and amorphous solids, regular and regular-like graphs "melt" at lower temperatures and display a sharper transition between connected to disconnected structures than the random spatial graphs, which resemble amorphous solids. Finally, we study computationally this transition in some real-world networks where we observe that the rate of melting of graphs depends on the topology of the corre-sponding network. In particular we observe that this rate changes either as an exponential or as a power-law with β. We also discover that the main driver for node melting is the eigenvector centrality of the corresponding node. That is, nodes with higher values of the Perron-Frobenious eigenvector melt at lower temperatures than those with smaller values of it.

Preliminaries
Here we shall present some definitions, notations, and properties associated with networks to make this work self-contained. We will use indistinctly the terms networks and graphs across the paper. Here we consider only simple, undirected graphs Γ = (V, E) with n nodes (vertices) and m edges. The notation used in the paper is the standard in network theory and the reader is referred to the monograph [9] for details. An important concept to be used across this paper is the one of walks. A walk of length k in Γ is a set of nodes path is a walk with no repeated nodes. A graph is connected if there is a path connecting every pair of nodes. Let A be the adjacency matrix of the graph Γ. For simple graphs A is symmetric and thus its eigenvalues are real, which we label here in non-increasing order: λ 1 ≥ λ 2 ≥ . . . ≥ λ n . We will consider the spectral decomposition of A: = U ΛU T , where Λ is a diagonal matrix containing the eigenvalues of A and U = [ − → ψ 1 , . . . , − → ψ n ] is orthogonal, where − → ψ i is an eigenvector associated with λ i . We consider here sets of orthonormalized eigenvectors of the adjacency matrix. Because the graphs considered here are connected, A is irreducible and from the Perron-Frobenius theorem we can deduce that λ 1 > λ 2 and that the leading eigenvector − → ψ 1 can be chosen such that its components ψ 1 (p) are positive for all p ∈ V . It is known that A k pq counts the number of walks of length k between p and q. The following result concerning the eigenvalue λ 2 is well-known in spectral graph theory.
Two important results that we will use in the current work are the following. For the sake simplicity let us suppose that the vertices of the graph Γ are labeled as V = {1, 2, 3, . . . , n}.
2. Some edge joins a vertex of P (ψ j ) to one of N (ψ j ), and P (ψ j ) ∪ N (ψ j ) , P (ψ j ) and N (ψ j ) are all connected. An important quantity for studying communication processes in networks has been defined as the communicability function [12,14].
Definition 5. Let p and q be two nodes of Γ. The communicability function between these two nodes is defined as It counts the total number of walks starting at node p and ending at node q, weighted in decreasing order of their length by a factor 1 k! ; therefore it is considering shorter walks more influential than longer ones. In this work we consider a generalization of the communicability function [13,11] consisting of where β ≥ 0 is a parameter that weights homogeneously every edge of the graph Γ. Let us give a complete physical interpretation of the communicability function by considering the following model. Let us consider a network of quantum-harmonic oscillators, such as every node is a ball of mass m and two nodes are connected by a spring of strength constant ω (see [13] for details). We tie the network to the ground (to avoid translational movement) with springs of constant K max k i as illustrated in Figure 2.1 (we remind that k i is the degree of the node i). The Hamiltonian describing the energy of this system is given bŷ where A ij are the elements of the adjacency matrix, a † i (a i ) are the annihilation (creation) operators, and Ω = K/mΩ.
Let us submerge the network of quantum harmonic oscillators into a thermal bath with inverse temperature β = (k B T ) −1 , where k B is a constant and T is the temperature. Then, the following result has been previously proved [13].
The thermal Green's function of the network of quantum harmonic oscillators described by 2.1 is Remark 7. The thermal Green's function accounts for how the node p (respectively q) 'feels' a perturbation at node q (resp. p) due to thermal fluctuations in the bath. When the temperature goes to infinity, the inverse temperature β → 0, which means that every edge in the graph vanishes and the resulting graph is trivial, similar to a system of free particles. When the temperature goes to zero, the inverse temperature β → ∞ indicating that an infinity number of edges are created between every pair of nodes connected in Γ ; a situation analogous to a rigid solid.

Melting phase transition
Let us consider a connected graph with eigenvalues ordered as λ 1 > λ 2 ≥ λ 3 ≥ ... ≥ λ n . Let us then write the communicability function in the following way where ψ + j (p) (ψ − j (p)) means that the pth entry of the jth eigenvector is positive (negative). Let us consider the graph illustrated in Figure 3 ∆G pq (β) = G pq (β) − e βλ 1 ψ 1 (p) ψ 1 (q), which can also be expressed as where the first term, which can be written as 2≤j≤n e βλ j ψ + corresponds to the case when both nodes have the same sign in the corresponding eigenvector, and the second term, which can be written as 2≤j≤n e βλ j ψ + accounts for the cases in which the two nodes have different sign in the corresponding eigenvector. We notice that the second term is always negative and we use the modulus of it to express the term ∆G pq (β) as a difference. ∆G pq (β) accounts for the difference between the in-and out-of-phase vibrations of the corresponding pair of nodes.
Let us now reconnect with Lindemanncriterion of melting [19] . According to Lindemann the average amplitude of thermal vibrations in crystals increases with the temperature up to a point in which the amplitude of vibration is so large that the atoms invade the space of their nearest neighbors and the melting starts. Lindemann criterion consists in considering that melting might be expected when the mean-square amplitude of vibrations exceed a certain threshold value [19]. Let us consider that in a graph Γ such threshold is given by That is, that melting starts in a given graph when the vibrations of the nodes p and q at a given temperature measured by ∆G pq (β) exceed the value of the maximum vibration of any pair of nodes in that graph at the same temperature, . We should notice that at a given temperature the terms ∆G pq (β) and M (Γ, β) may be either positive or negative. Thus, in order to implement the Lindemann criterion on graphs we should sum both terms instead of having their difference, Then, when M (Γ, β) > 0 we have the following scenarios. If ∆G pq (β) > 0 then ∆G pq (β) is always positive as it is the sum of two positive terms indicating a reinforcement of the inphase vibrations of the two nodes. If ∆G pq (β) < 0 then ∆G pq (β) > 0 if the difference between the in-phase and out-of-phase vibrations ∆G pq (β), does not overtake the maximum in-phase vibrations of any pair of nodes in the graph. Otherwise, ∆G pq (β) < 0, which indicates that the out-of-phase vibrations of these two nodes have overtaken not only their in-phase vibrations but also the maximum in-phase vibrations of any pair of nodes in the graph. In this last case we will say that the corresponding edge has been melted. On the other hand, when M (Γ, β) < 0, then also ∆G pq (β) < 0 which means that ∆G pq (β) < 0 and the edge necessarily melts. Then, we will call ∆G pq (β) the graph Lindemann criterion, having in mind that the melting will start when ∆G pq (β) < 0. Let us define the following representation of ∆G pq (β) in the form of a new graph.
is the graph with the same set of nodes as Γ and with edge set given by the following adjacency relation Definition 9. Let Γ = (V, E) be a simple graph and let H (V, E , β) be its communicability graph. Let p and q be two nodes of Γ. We say that there is a Lindemann path between the nodes p and q in Γ at a given value of β if there is a path connecting both nodes in the communicability graph H (V, E , β). In this case we say that ∃L p,q . Otherwise, we say that We now define a graph that contain all the information about the in-and out-phase nature of the vibrations in a graph Γ = (V, E).
Definition 10. Let Γ = (V, E) be a simple graph and let H (V, E , β) be its communicability graph. The Lindemann graph F (V, E , β) of Γ is the graph with the same set of vertices as Γ and edge set defined by the following adjacency relation

Main result
Here we prove the existence of the phase transition between connected to disconnected Lindemann graph for any simple graph. For proving this result it is enough to prove that this transition occurs in the communicability graph. Then, when the communicability graph is connected also is the Lindemann graph, due to the fact that there is a path between every pair of nodes in a connected graph. In the same way, if the communicability graph is disconnected also is the Lindemann graph because there will be pairs of adjacent nodes of the original graph for which there are no paths connecting them in the communicability graph. We divide our result into two parts. The first deals with all graphs which are not complete multipartite ones.
The second proves the result for this kind of graphs.
Theorem 11. Let H(Γ, β) be the communicability graph for a non complete multipartite graph Proof. We start by proving that the communicability graph H(Γ, β) is disconnected for certain value of the inverse temperature. Then, we prove that it becomes connected for certain value of β, which immediately implies that the communicability graph H(Γ, β) makes a transition from connected to disconnected at certain intermediate temperature, which we call β c . This value is unique since the communicability function is monotonic, as it is the sum of exponential functions which are monotonic. The communicability graph function is Let us consider the case when β → 0. In this case we have lim β→0 Then, it is obvious that the edge pq in the graph is disconnected and this happens for every pair of nodes in the graph. Consequently, the communicability graph is disconnected for some β when β → 0.
If ∩ r j=2 O(ψ j ) = ∅, then by Lemma 2, one of the following two separate cases hold. Then ψ j (p) ψ j (q) ≥ 0, ∀j ∈ {2, 3, . . . , r}, and ∀(p, q) ∈ E, G pq (4.3) satisfies: Now let us rewrite the function G pq in the following form, where s and t are two distinct nodes in the graph: Let us now consider β = 0, then ∀p = q ∈ V, Case 2. There exists at least one edge that joins a vertex of P(ψ j ) to one of N (ψ j ), moreover P(ψ j ) and N (ψ j ) are connected.
Remark 12. In the case of complete multipartite graphs which were not included in the Theorem 11 we have the following. As in the general case ∆G pq (β → 0) < 0, which indicates that the edge does not exist, and this is true for any pair of nodes in the graph. Then, we have to show when such edge exists. According to Lemma 1 [29] in complete multipartite graphs λ 2 ≤ 0. Then, let us first consider the case when λ 2 = 0. In this case when β → ∞, the communicability function G pq (β) is dominated by the term of the largest eigenvalue λ 2 .
Then G pq (4.9) satisfies G pq (β) ≥ 0. So let us denote the subgraphs generated by W h , and the other nodes of V which do not belong to W h , by W 1 , W 2 , · · · , W 2 r−1 . Finally, in the same previous way we can connect W 1 , W 2 , · · · , W 2 r−1 to each other, such that we get a connected graph. Let p ∈ W h and q ∈ W s , where h, s ∈ {1, 2, . . . , 2 r−1 }, such that the absolute value of r j=2 ψ j (p ) ψ j (q ), satisfies: Then G pq (4.9) satisfies Therefore W 1 , W 2 , · · · , W 2 r−1 are connected to each other. Then from 4.5, 4.10 and 4.11 On the other hand, when λ 2 < 0 we have that This means that lim β→∞ ∆G pq (β) = 0. However, such limit can be either positive-existence of the edge-or negative-not existence of the edge. Thus, in this case we can consider that the edge exists if lim β→∞ ∆G pq (β) ≤ ε, where ε is a threshold very close to zero. The case of complete multipartite graphs is an all-or-nothing case of melting. That is, at β = 0 all the nodes in the Lindemann graph are isolated. When β is very large all the nodes in the communicability graph are connected to each other and the Lindemann graph is the same as the original graph. Thus, in these graphs there is not a gradual melting as in the rest of the graphs but an abrupt transition between being connected to fully disconnected occurs.

Melting of granular materials
In this section we study the influence of order and randomness on the melting phase transition in graphs. The influence of order vs. randomness is on the basis of many physical problems. In particular, here we are interested in using graphs as a model of solids and granular materials with ordered structures vs. those having a random one. The classical example of an ordered system is a crystal where atoms or molecules are arranged in a repeating pattern [31]. On the other hand, amorphous solids, which are characterized by the lack of regular pattern or repetition, are good examples of random-like materials [1]. In order to model a random-like material we consider here a type of random graph known as the Gabriel graph [15]. The The reason why we consider random neighborhood (Gabriel) graphs here instead of other types of random graphs is the following. To keep the analogy with solid materials we should maintain certain geometric disposition of the nodes. This geometric arrangements of nodes are possible in the so-called random geometric graphs (RGGs) as well as in the random neighborhood graphs (RNGs). The RGGs are nonplanar graphs, which implies that node A can interact with another B even in the case that a third node C is exactly in the middle between A and B. This, of course, is not a realistic scenario for the interaction between atoms or molecules and not appropriate for representing a solid. However, Gabriel graphs are planar graphs and avoid the interaction between mutually occluded nodes. Consequently, they are appropriate to model amorphous solids.
The differences in the ordered vs. random arrangement of atoms/molecules in crystalline and amorphous solids make that they differ significantly in the way they change their phase Another possibility of the current approach is that it allows us to visualize the evolution of the "melting" process in graphs in order to gain insights about its mechanism. In At very low values of β, e.g. β = 0.000025 (Fig. 5.3 (c)) the communicability structure of the lattice resembles a trivial graph in which almost every node is isolated. As the temperature drops, β increases, certain structures start to emerge. In particular, for β = 0.0005 (Fig. 5.3 (b)) an annulus-external part of the lattice-is solidified into a single connected component and only the central part of the graph remains melted. As the temperature drops below β = 0.000075 (Fig. 5.3 (a)), the melted region-red nodes-shrinks to the very center of the lattice. The observed pattern of melting of the square lattice is similar to the one observed experimentally for crystalline solids. In Fig. (Fig. 5.3 (d)) we illustrate the results of Wang et al. [33] for the melting of colloidal crystals which show such patter of central melting.   In the case of the amorphous graph there is no repeating pattern in them, and it is impossible to find a general structural pattern of the evolution of the melting process. A few snapshots of the process are given in Fig. 5.4. The temperature needed to melt these graphs is significantly higher-smaller β-than the ones needed to melt square lattices of the same size, which coincides with our previous observations as well as with the experimental results for crystalline and amorphous solids. The reasons of this significant difference will become clear in the next section of this work.

Complex networks
The term complex networks is frequently used to refer to graphs representing the skeleton of complex systems, such as social, ecological, infrastructural, technological and biomolecular networks [9]. Here we consider a series of 47 complex networks-see Supplementary Information for description and references-arising from these different scenarios. Here we split our analysis into two parts. First we consider global properties of the networks and then we analyze the influence of node-level centrality on the melting process of these networks.

Global analysis
We start here by finding the value of β at which the transition between connected to discon- The correlations found for ln β c with some of the previous structural parameters may be hiding something about the real structural characteristic of networks that influence their "melting". For instance, the negative correlation between edge density and β c seems suspicious. Our intuition tells us that, under all other structural conditions the same, high density networks should melt at higher temperatures, i.e., lower β c , than lower density ones. This is exactly what it is observed in molecular crystals of nonpolar molecules, such as linear alkanes [3]. In addition, in the previous section we have studied two different kinds of networks which differ very significantly in theirs values of β c in spite of the fact that they have exactly the same number of nodes and edge densities. While the square lattice is an almost regular graph, in a Gabriel graph there are small degree heterogeneities that emerge from the clustering of groups of nodes in a relatively close space. Then, the fact that smaller and denser (real-world) networks are the ones having the largest β c , i.e., they have Lindemann graphs easier to dis-connect, may indicate that the "degree homogeneity" of these networks more than their sizes or densities is the real driver of their melting. In order to capture these degree irregularities we recall the definition of the average degree of a network The right-hand side of the previous equation is useful to think that the spectral radius of the adjacency matrix is a sort of average degree, which instead of counting only the number of nearest neighbors of a node consider also a more global picture around it Notice thatk ≤ λ 1 with equality if and only if the graph is regular. Thus, the term λ 1 /k represents the ratio of a more global environment of a node to its more local one. That is, the ratio λ 1 /k indicates how a node "sees" as average its global environment in relation to its nearest neighbors. In a regular graph its local environment, i.e., its degree, is identical to the degree of its neighbors, second neighbors, and so on and we get that λ 1 /k =1. Then, we can define the following index of global to local degree heterogeneity Notice that (G) = λ 1 δ , which may explain the previously observed correlation between ln β c and ln δ. between global and local degree heterogeneities. Regular graphs are easier to melt than nonregular ones, and the more irregular-in terms of global to local degree heterogeneity-the graph is the smallest the value of β c , i.e., more difficult to melt.

Local analysis
In this subsection we are interested in the local analysis of the effects of decreasing the value of β on the topological structure of a network. In particular we investigate computationally two important aspects of the graph melting process: (i) How the nodes of a network melt?
and (ii) Which structural parameter drives the melting of the nodes? For investigating these questions we consider a subset of the real-world networks studied in this work. We create a melting barcode plot in which we plot every node in the y-axis and in the x-axis we provide the value of β at which the corresponding node disconnect from the giant connected component of the graph. In Fig. 5.6 we illustrate the melting barcodes of three networks: neurons (a), Little Rock (b) and corporate elite (c). We need to read these melting barcodes from right to left as the melting process starts at higher values of β and proceed by decreasing it. There are significant differences in the three barcodes presented which point out to the differences existing in the melting processes of the different graphs analyzed. First, we can observe that the shape of the melting barcodes are different. While in "neurons" the decay resembles an exponential curve, in "Little Rock" it is almost linear and in the "corporate elite" it displays a more skewed shape (see further for quantitative analysis). In the second place, the barcodes of "Little Rock" and of corporate elite display regions in which large groups of nodes are disconnected at the same temperature, while in neurons the change is smoother.
We then investigate the rate of change of the melting process in the networks analyzed by considering the shape of the histogram of the number of nodes "melted" at a given temperature.
That is, we construct the histograms of the number of nodes melted in a temperature range versus the range of temperatures. In general we observe two kinds of decay of the number of nodes melted at a given temperature in relation to the inverse temperature. They are: where η is the number of nodes melted at a given value of β. For some of the smallest networks it was not possible to find any particular law of the decay of η as a function of β. These were the cases of the networks of Benguela (n = 29), Coachella (n = 30), Social3 (n = 32), St.
Marks (n = 48), as well as for the network of Little Rock, which is not so small (n = 181) but it also has a very disperse histogram. For the rest of the networks analyzed we display the parameters of the fitting to Eqs. 5.4 and 5.5 in Table 1.
The fitting parameters given in Table 1  Finally, we investigate which structural parameters determine the melting process of the nodes of a network. In particular we consider here the role of node centrality on the melting of the corresponding node. We then analyze the relation between the value of β at which a node melts and its degree (DC), closeness (CC), betweenness (BC), eigenvector (EC) and subgraph centrality (SC). All these measures are defined in the Supplementary Information.
In general, we observe that the values of β at which the nodes melt correlate very well with EC.
All networks studied displayed Pearson correlation coefficients between these two parameters higher than 0.90, with the exceptions of the networks of Benguela and Macaque visual cortex.
In addition we investigate the coefficient of variation (CV) of the values of β at which a node melts estimated from a linear regression with EC. This coefficient is given by the standard In general, we observe that when β c is arbitrarily small, the correlation between the value of β at which the node melts and EC is better than when β c is relatively large, e.g. Benguela, Coachela, Social3, Macaque, St. Marks, etc. The reason for that difference is the following.
Let us recall that at β c the value of the Lindemann criterion is negative, that is ∆G pq (β c ) = n j=2 ψ j (p) ψ j (q) e βλ j < 0.

Conclusions and future outlook
The most important result of the current work is the proof of the existence of a universal melting transition in graphs and networks. This transition takes place when we consider a Lindemann-like model on graphs, which is based on a vibrational approach of nodes and edges.
From a mathematical point of view the current method is based on the spectral properties of the adjacency matrix of the graph and the changes taking place on the exponential matrix function exp (βA) with the changes of β. In this way, regular-like graphs like square lattices are easier to melt than more irregular structures, such as spatial planar graphs. These differences resemble the known dissimilarities between crystalline and amorphous solids in their melting.
The analysis of a series of real-world networks has given us the possibility of exploring the global and local structural characteristics of networks which drives their melting. At the global topological level, we have shown here that the value β c at which the melting of a graph occurs depends mainly on the differences between the local and global degree heterogeneities existing in the graph. At the local one we have observed that the melting is triggered by the nodes having the higher eigenvector centrality in the network, particularly in those cases where the melting temperature is very close to zero.
The analysis of graph/networks melting as proposed here opens many new possibilities for the study of network robustness to external stresses. There are many mathematical and computational questions that remain open from the current study. They include, but are not limited, to the following ones: (i) A more exhaustive analysis of the topological (global and local) drivers of the graph melting; (ii) How certain specific network characteristics, e.g., clustering, modularity, degree assortativity, etc., influence the melting temperature and the melting process of artificial and real-world networks?; (iii) Is there a ranking of certain classes of graphs, e.g., trees, monocyclic graphs, etc., according to their melting? We hope the reader can help to answer some of these questions and generate new ones that clarify our understanding of network robustness to external stresses.