State-Space Network Topology Identification From Partial Observations

Abstract

In this work, we explore the state-space formulation of network processes to recover the underlying structure of the network (local connections). To do so, we employ subspace techniques borrowed from system identification literature and extend them to the network topology inference problem. This approach provides a unified view of the traditional network control theory and signal processing on networks. In addition, it provides theoretical guarantees for the recovery of the topological structure of a deterministic linear dynamical system from input-output observations even though the input and state evolution networks can be different.Index Terms-state-space models, topology identification, graph signal processing, signal processing over networks INTRODUCTIONIn recent years, major efforts have been focused on extend traditional tools from signal processing to cases where the acquired data is not defined over typical domains such as time or space but over a network (graph) [1,2]. The main reason for the increase of research in this area is due to the fact that network-supported signals can model complex processes. For example, by means of signals supported on graphs we are able to model transportation networks [3], brain activity [4], and epidemic diffusions or gene regulatory networks [5], to name a few.As modern signal processing techniques take into account the network structure to provide signal estimators [6-8], filters [9-12], or detectors [13][14][15], appropriate knowledge of the interconnections of the network is required. In many instances, the knowledge of the network structure is given and can be used to enhance traditional signal processing algorithms. However, in other cases, the network information is unknown and needs to be estimated. As the importance of studying such structures in the data has been noticed, retrieving the topology of the network has become a topic of extensive research [16][17][18][19][20][21][22][23][24].Despite the extensive research done so far (for a comprehensive review the reader is referred to [2,17] and references therein), most of the approaches do not lever a physical model beyond the one induced by the so-called graph filters [18,25] drawn from graph signal processing (GSP) [10,26,27]. Among the ones that propose a different interaction model e.g., [20,24], none of them considers the network data (a.k.a. graph signals) as states of an underlying process nor considers the that the input and the state may evolve on different underlying networks. However, different physical systems of practical interest can be defined through a state-space formulation with

Structure Identification of Fractional-Order Dynamical Network with Different Orders

“…In this paper, all the state variables are assumed to be observable. In real cases, the state variables may be partially observable [11]. Thus, how to identify the topology from partial observations for fractional-order networks, especially with different orders, deserves further studies.…”

“…Since the DNA interactions have a great effect on cellular processes, how to effectively identify them is an interesting and important issue and deserves deep study. Thus far, researchers have performed much research on the structure identification issue of dynamical networks [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. In [15], Waarde et al addressed the problem of identifying the graph structure of a dynamical network using measured input/output data.…”

Learning Bollobás-Riordan Graphs Under Partial Observability

“…However, all the aforementioned works do not consider the time dynamics of the signals emitted by the nodes, and, hence, they are not applicable to dynamical systems like the VAR model considered here. For dynamical graph models, relevant results under full observability were presented in [13][14][15][16][17][18][19], whereas partial observability was recently addressed in [20][21][22][23][24][25]. Particularly relevant to our work is the setting considered in [23][24][25], where the VAR model (1) runs on top of an Erdős-Rényi random graph [26,27].…”

Learning connectivity and higher-order interactions in radial distribution grids

“…To capture the nonlinear connectivity and dynamics in the data, topology identification methods combining partial correlations and kernels have recently been investigated [19], [9], [6]. While it is possible to adopt these methods to identify the distribution grid topology, they face two challenges in practice: First, selecting proper kernels requires cross-validations, or solving computationally involved optimization tasks.…”