We consider a class of weakly interacting particle systems of mean-field type. The interactions between the particles are encoded in a graph sequence, i.e., two particles are interacting if and only if they are connected in the underlying graph. We establish a Law of Large Numbers for the empirical measure of the system that holds whenever the graph sequence is convergent in the sense of graph limits theory. The limit is shown to be the solution to a non-linear Fokker-Planck equation weighted by the (possibly random) graph limit. No regularity assumptions are made on the graphon limit so that our analysis allows for very general graph sequences, such as exchangeable random graphs. For these, we also prove a propagation of chaos result. Finally, we fully characterize the graph sequences for which the associated empirical measure converges to the mean-field limit, i.e., to the solution of the classical McKean-Vlasov equation.

pdf/weakly-interacting-oscillators-on-dense-random-graphs-o13gl8gizq.pdf

Weakly interacting oscillators on dense random graphs

OUP accepted manuscript

Inhomogeneous random K-out graphs were recently introduced to model heterogeneous sensor networks secured by random pairwise key predistribution schemes. First, each of the $n$ nodes is classified as type-1 (respectively, type-2) with probability $0 (respectively, $1-\mu$ ) independently from each other. Next, each type-1 (respectively, type-2) node draws 1 arc towards a node (respectively, $K_{n}$ arcs towards $K_{n}$ distinct nodes) selected uniformly at random, and then the orientation of the arcs is ignored. It was recently established that the inhomogeneous random K-out graph is 1-connected asymptotically almost surely (a.a.s.) if and only if $K_{n}=\omega (1)$ . In this work, we analyze the $k$ -connectivity of inhomogeneous random K-out graphs; i.e., with $k=1, 2, \ldots $ , the property that the network remains connected despite the removal of any $k-1$ nodes or links. We first establish a zero-one law for the property that the minimum node degree is at least $k$ . In particular, we present scaling conditions on $\mu $ and $K_{n}$ such that the resulting graph has minimum degree at least $k$ with probability approaching one (respectively, zero) constituting the one-law (respectively, zero-law), as the number of nodes gets large. We show that for $k=2, 3, \ldots $ , we need to set $K_{n} = \frac {1}{1-\mu }(\log n +(k-2)\log \log n + \omega (1))$ for the network to have a minimum node degree of at least $k$ a.a.s. Next, we prove that having $K_{n} = \frac {1}{1-\mu }(\log n +(k-2)\log \log n + \omega (1))$ also ensures that the graph is $k$ -connected a.a.s., meaning that the zero-one laws for minimum node degree and $k$ -connectivity coincide. We present simulation results to demonstrate the usefulness of the results in the finite node regime. The results given here indicate an interesting fact about inhomogeneous random K-out graphs, i.e., that the number of additional edges needed to go from 1-connectivity to $k$ -connectivity with $k \geq 2$ is unexpectedly larger as compared to many classical random graph models studied before.

On the Minimum Node Degree and k-Connectivity in Inhomogeneous Random K-Out Graphs

In network data analysis, one of the fundamental data mining tasks is to find dense subgraphs, which has wide applications in biology, finance, graph compression and so on. In practice, given netwo...

Test dense subgraphs in sparse uniform hypergraph

Recent discovery of the extraordinary-log universality has generated tremendous interest in classical and quantum boundary critical phenomena. However, the classical-quantum correspondence of such a universality remains elusive. Here, by utilizing quantum Monte Carlo simulations, we study the quantum edge criticality of a two-dimensional Bose-Hubbard model featuring emergent bulk criticality. On top of an insulating bulk, the open edges experience a Kosterlitz-Thouless-like transition into superﬂuid phase when sufﬁciently enhancing the hopping strength on edges. At the bulk critical point, the edges exhibit the special, the ordinary and the extraordinary critical phases. In the extraordinary phase, logarithms are involved in the ﬁnite-size scaling of two-point correlation and superﬂuid stiffness, which admit a classical-quantum correspondence for the extraordinary-log universality. Thanks to the modern quantum emulator for interacting bosons in lattices, the edge critical phases might be realized in experiments.

Quantum extraordinary-log universality of boundary critical behavior

We study the asymptotic behavior of the clique number in rank-1 inhomogeneous random graphs, where edge probabilities between vertices are roughly proportional to the product of their vertex weights. We show that the clique number is concentrated on at most two consecutive integers, for which we provide an expression. Interestingly, the order of the clique number is primarily determined by the overall edge density, with the inhomogeneity only affecting multiplicative constants or adding at most a $\log\log(n)$ multiplicative factor. For sparse enough graphs the clique number is always bounded and the effect of inhomogeneity completely vanishes.

Cliques in rank-1 random graphs: the role of inhomogeneity.

We study the asymptotic behavior of the clique number in rank-1 inhomogeneous random graphs, where edge probabilities between vertices are roughly proportional to the product of their vertex weights. We show that the clique number is concentrated on at most two consecutive integers, for which we provide an expression. Interestingly, the order of the clique number is primarily determined by the overall edge density, with the inhomogeneity only affecting multiplicative constants or adding at most a $\log \log (n)$ multiplicative factor. For sparse enough graphs the clique number is always bounded and the effect of inhomogeneity completely vanishes.

Cliques in rank-1 random graphs: The role of inhomogeneity

We formalize the problem of detecting the presence of a botnet in a network as an hypothesis testing problem where we observe a single instance of a graph. The null hypothesis, corresponding to the absence of a botnet, is modeled as a random geometric graph where every vertex is assigned a location on a $d$-dimensional torus and two vertices are connected when their distance is smaller than a certain threshold. The alternative hypothesis is similar, except that there is a small number of vertices, called the botnet, that ignore this geometric structure and simply connect randomly to every other vertex with a prescribed probability. 
We present two tests that are able to detect the presence of such a botnet. The first test is based on the idea that botnet vertices tend to form large isolated stars that are not present under the null hypothesis. The second test uses the average graph distance, which becomes significantly shorter under the alternative hypothesis. We show that both these tests are asymptotically optimal. However, numerical simulations show that the isolated star test performs significantly better than the average distance test on networks of moderate size. Finally, we construct a robust scheme based on the isolated star test that is also able to identify the vertices in the botnet.

pdf/detecting-a-botnet-in-a-network-56bcofpezr.pdf

Detecting a botnet in a network

We study the problem of detecting whether an inhomogeneous random graph contains a planted community. Specifically, we observe a single realization of a graph. Under the null hypothesis, this graph is a sample from an inhomogeneous random graph, whereas under the alternative, there exists a small subgraph where the edge probabilities are increased by a multiplicative scaling factor. We present a scan test that is able to detect the presence of such a planted community, even when this community is very small and the underlying graph is inhomogeneous. We also derive an information theoretic lower bound for this problem which shows that in some regimes the scan test is almost asymptotically optimal. We illustrate our results through examples and numerical experiments.

Community detection in inhomogeneous random graphs

Detecting a planted community in an inhomogeneous random graph.

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